AC servo motors, In-Wheel motors, hub motors, Gear motors,Frameless Brushless motors, Inner rotor Brushless motors, Outer rotor brushless rotors.
Discovery of motors 
Here, you can see what companies have what kind of motors and what sizes of motors from below link. 
The forcus is on mainly Japanese players but there are some global players too.
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What kind of motors are in the world?
It is difficult to classify between AC syncronous servo motor and DC brushless motor with encoder clearly because the structure of the motor part is same. It can be classified based on type of driver(amps) or just depending on how to call by manufacturers.
 Although some companies have own unique technology to maximize higher torque density in the limited space inside of motors and therfore they have robust presence in the specific market, generally a motor performance depends on the volume of motor body, which means if the size of motor and a kind of motor are same, achievable motor performance in terms of torque density is same. And a bigger motor can achieve higher performance. So, if you know who have what kind of motor and what size of motor, such inputs help you when you look for motors you need. 
Also, especially for small motors, customizing will be chosen in many of cases. But when we say "customization", the meaning is wide. It includes from changing in only length of cables, winding change to full customization which makes motors having preferable size and performance by investing all of tooling. In many cases, customers must want to choose standard or small custom products because of saving a cost. When you think about a custom motor, the point is whether the manufacturer has its size(diameter(OD) or square) because if the manufacturer has the motors corresponding the diameter(OD) you look for, even if the length is different, the cost could minimize in some cases because they may be able to use some equipment, parts of tooling and some machine they original have. Above information will be for your reference when you look for a motor.

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